Unitary Operators In The Orthogonal Complement Of A Type Ⅰ Von Neumann Algebra In A Type Ⅱ₁Factor

It is well-known that the equality LG(?)LH=span{Lg:g∈G-H holds for G an i.c.c. group and H a subgroup in G, where LG and LH are the corresponding group von Neumann algebras, and LG(?) LH is the set{x€LG:EL (x)=0} with ELH the conditional expectation defined from LG onto LH. Inspired by this, it is natural to ask whether the equality N(?)A=span{u:u is unitary in N (?) A}SOT holds for N a type Ⅱ1factor and A a von Neumann subalgebra of N. In this paper, we give an affirmative answer to this question for the case A a type I von Neumann algebra.